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The Segal Algebra \({\bf S}_0({\Bbb R}^d)\) and Norm Summability of Fourier Series and Fourier Transforms

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Abstract.

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L 1-norm convergence of the θ-means σ n θ f to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger’s Segal algebra \({\bf S}_0({\Bbb R}^d)\), then these convergence results hold. Some new sufficient conditions are given for θ to be in \({\bf S}_0({\Bbb R}^d)\). A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case.

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Authors and Affiliations

  1. University of Vienna, Austria

    Hans G. Feichtinger

  2. Eötvös L. University, Budapest, Hungary

    Ferenc Weisz

Authors
  1. Hans G. Feichtinger
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  2. Ferenc Weisz
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Additional information

This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.

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Feichtinger, H., Weisz, F. The Segal Algebra \({\bf S}_0({\Bbb R}^d)\) and Norm Summability of Fourier Series and Fourier Transforms. Mh Math 148, 333–349 (2006). https://doi.org/10.1007/s00605-005-0358-4

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  • DOI: https://doi.org/10.1007/s00605-005-0358-4

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